The inventors have previously described a technique for measuring the conductivity and sheet resistance of thin-film samples of graphite comprising monolayers and very few layers of graphene in their paper “Non-contact method for measurement of the microwave conductivity of graphene” in Applied Physics Letters, Vol. 103, pp. 123103-1-123103-4 (2013). This technique uses a high Q-factor dielectric resonator, made of a single crystal of a high permittivity material with a low microwave loss tangent, such as sapphire, which is excited by microwaves in a conducting chamber, to measure the conductivity of such thin-film samples in a non-contacting manner. As described at the end of that paper, the present inventors indicated that it would also be desirable to find a technique for measuring the carrier mobility of thin-film samples in a similarly non-contacting way.
Until now, the carrier mobility of thin-film samples such as monolayers and very few layers of graphene has been measured using a contacting method, which comprises patterning a thin-film sample into a Hall bar device on a substrate, such as an oxidised silicon wafer, and attaching electrodes to the sample in order to determine the carrier mobility of the sample from field-effect and magneto resistance measurements, as described, for example, in “Electrical Field Effect in Atomically Thin Carbon Films” by K. S. Novoselov, A. K. Geim et al. in Science, Vol. 306, pp. 666-669 (22 Oct. 2004) and in “Two-dimensional atomic crystals” by K. S. Novoselov et al. in Proceedings Nat. Acad. Sci., Vol. 102, no. 30, pp. 10451-10453 (26 Jul. 2005).
Such techniques can exhibit the disadvantages of being time-consuming and inefficient to prepare the samples as Hall bar devices and destructive of the thin-film sample to be measured.
On the other hand, it is also known to measure the carrier mobility of a semiconductor sample in a non-contacting way by placing such a sample in a rectangular cavity resonator and injecting the cavity with microwaves to excite the sample into an orthogonal mode in the presence of a static magnetic field. However, this technique has the disadvantage that although the sample is not destroyed by such a technique, it is difficult to characterize the Hall coefficient of the sample and its mobility with any accuracy. This is for several reasons. Firstly, it is difficult to achieve a high Q-factor in the cavity. Secondly, only samples of very small volume can be used and usually the shape and position of the sample within the cavity are critical, since the complicated geometry of the rectangular cavity relative to the sample makes calculation of the coupling between the sample and the cavity problematic. Furthermore, there is a direct effect of the magnetic field on the conducting walls of the cavity, which arises from the small but finite Hall coefficient of the metal from which the cavity is made. This technique is therefore not favoured.
Reference is also made to the article Measurement Science and Technology, Vol. 24, 2013, Jerzy Krupka, “Topical Review; Contactless methods of conductivity and sheet resistance measurement for semiconductors, conductors and superconductors; Contactless methods of conductivity and sheet resistance measurement for semiconductors, conductor”. This discusses microwave techniques in Section 5 and microwave measurements of charge carrier mobility and charge carrier concentration in Section 6.